## Relations and Functions

# Types of Relations

- Identity Relation: R is an identity relation if (a, b) ∈ R if a = b, a ∈ A, b ∈ A. In other words, every element of ‘A’ is related to only itself.
- Universal Relation: Let ‘A’ be any set and ‘R’ be the set A × A, then R is called the universal relation in ‘A’.
- Void Relation: φ is called void relation in a set.
- Types of Relations:

Reflexive: If aRa ∀ a ∈ A i.e. if (a, a) ∈ R ∀ a ∈ A

Symmetric: If aRb ⇒ bRa ∀ a, b ∈ A, i.e. if (a, b) ∈ R ⇒ (b, a) ∈ R

Antisymmetric: If aRb and bRa ⇒ a = b, ∀ a, b ∈ A

Transitive: If aRb and bRc ⇒ aRc, ∀ a, b, c ∈ A, i.e. (a, b) ∈ R and (b, a) ∈ R ⇒ (a, c) ∈ R, ∀ a, b, c ∈ A

Equivalence relation: It should be reflexive, symmetric and transitive. - Inverse relation: If R is a relation from A to B, then R
^{−1}is a relation from B to A and is defined as R^{−1}= {(b, a) / (a, b) ∈ R}

i.e. aRb ⇒ bR^{−1}a

Domain of R = Range of R^{−1}Range of R = Domain of R^{−1} - Composite relation: If R ⊆ A × B and S ⊆ B × C then the composite relation is SoR ⊆ A × C

SoR = {(a, b) ∈ R and (b, c) ∈ S ⇒ (a, c) ∈ SoR} - Partially ordered relation: A relation R is defined on the set A is said to be partially ordered relation if it is reflexive and antisymmetric and transitive.
- Every identity relation is reflexive, converse need not be true also identity relation is reflexive, symmetric and transitive
- The relation R on a set A is symmetric if and only if R = R
^{−1}. - The identity relation on a set is an anti-symmetric relation.
- A relation which is not symmetric need not be antisymmetric.
- a and b are relatively prime means the greatest common divisor of a and b is equal to 1
- a ≡ b (mod m) means a – b is divisible by m.

### Part1: View the Topic in this video From 48:30 To 52:54

### Part2: View the Topic in this video From 00:40 To 54:40

### Part3: View the Topic in this video From 00:40 To 15:48

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- The number of reflexive relations from A to A is 2
^{n(n − 1)}. If n(A) = n. - The number of symmetric relations from A to A is 2^{\frac{n(n+1)}{2}} if n(A) = n.